Simplified Design of Thin-Film Polarizing Beam Splitter Using Embedded Symmetric Trilayer Stack

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Electrical Engineering Faculty Publications Department of Electrical Engineering

7-1-2011

Simplified design of thin-film polarizing beam splitter using embedded symmetric trilayer stack

R. M.A. Azzam University of New Orleans, [email protected]

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Recommended Citation R. M. A. Azzam, "Simplified design of thin-film polarizing beam splitter using embedded symmetric trilayer stack," Appl. Opt. 50, 3316-3320 (2011)

This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact [email protected]. Simplified design of thin-film polarizing beam splitter using embedded symmetric trilayer stack

R. M. A. Azzam Department of Electrical Engineering, University of New Orleans, New Orleans, Louisiana 70148, USA ([email protected])

Received 28 March 2011; accepted 6 May 2011; posted 17 May 2011 (Doc. ID 144834); published 1 July 2011

An analytically tractable design procedure is presented for a polarizing beam splitter (PBS) that uses frustrated total internal reflection and optical tunneling by a symmetric LHL trilayer thin-film stack embedded in a high-index prism. Considerable simplification arises when the refractive index of the high- index center layer H matches the refractive index of the prism and its thickness is quarter-wave. This leads to a cube design in which zero reflection for the p polarization is achieved at a 45° angle of incidence independent of the thicknesses of the identical symmetric low-index tunnel layers L and L. Arbitrarily high reflectance for the s polarization is obtained at subwavelength thicknesses of the tunnel layers. This is illustrated by an IR Si-cube PBS that uses an embedded ZnS–Si–ZnS trilayer stack. © 2011 Optical Society of America OCIS codes: 230.1360, 230.5440, 240.0310, 260.5430, 310.6860.

1. Introduction the low-index tunnel layers and is propagating in Thin-film polarizing beam splitters (PBSs) are versa- the prism and center layer. In Sections 2 and 3 we tile optical elements that use destructive interfer- consider an important and analytically tractable spe- ence of light for one linear polarization (p or s) and cial case in which the refractive index of the center nearly full constructive interference for the ortho- layer is deliberately matched to that of the prism, gonal polarization in a multilayer stack at oblique i.e., n2 ¼ n0. This simple design employs only two incidence [1–10]. optical materials and depends on fewer design pa- This paper follows up on previous work by Azzam rameters, namely only one index ratio N ¼ n0=n1, and Perla [11,12] that dealt with the polarizing prop- angle of incidence ϕ0, and film thicknesses d1, d2. erties of a symmetric LHL trilayer stack (that con- A novel PBS cube (ϕ0 ¼ 45°)p isffiffiffi introduced in Section 4 1=2 sists of a high-index center-layer H of refractive that uses index ratio N ¼ð 2 þ 1Þ ¼ 1:55377 and index n2 and thickness d2 sandwiched between two center layer of quarter-wave optical thickness. This identical low-index tunnel layers L of refractive PBS, which uses tunnel layers of subwavelength index n1 and thickness d1), which is embedded in thicknesses, fully transmits the p polarization and a prism of high refractive index n0 > n1, Fig. 1. All nearly totally reflects the s polarization. Section 5 media are considered to be transparent and optically provides a brief summary of this paper. isotropic and are separated by parallel-plane bound- aries. Light is incident from medium 0 at an angle 2. Constraint on Film Thicknesses for Zero Reflection ϕ0 > the critical angle ϕc01 ¼ arcsinðn1=n0Þ of the 01 of the p or s Polarization interface, so that frustrated total internal reflection In [11] it was shown that the condition of zero reflec- (FTIR) and optical tunneling through the trilayer tion of the ν polarization (ν ¼ p, s) by an embedded stack take place. The optical field is evanescent in symmetric trilayer stack can be put in the form ℓ þ mX − nX2 X ¼ 1 1 : ð Þ 0003-6935/11/193316-05$15.00/0 2 2 1 © 2011 Optical Society of America −n þ mX1 þ ℓX1

3316 APPLIED OPTICS / Vol. 50, No. 19 / 1 July 2011 Fig. 1. Embedded symmetric trilayer stack as a PBS operating under conditions of FTIR. p and s are the linear polarizations parallel and perpendicular to the plane of incidence, respectively, and ϕ0 is the angle of incidence.

When index matching (n2 ¼ n0) is satisfied, ℓ, m,and citly on the index ratio N and angle of incidence n in Eq. (1) are determined by only one Fresnel ϕ0 via the 01-interface reflection phase shift δ. reflection coefficient r01ν at the 01 interface: From the round-trip phase thickness θ2 of the high-index center layer, the corresponding metric 2 3 ℓ ¼ r01ν; m ¼ −r01νð1 þ r01νÞ; n ¼ −r01ν: ð2Þ thickness d2 is determined by

−1 Upon substitution of ℓ, m, and n from Eqs. (2)in d2=λ ¼½ð4πn0Þ sec ϕ0θ2: ð7Þ Eq. (1), the condition of zero reflection of the ν polarization reduces to Likewise, the metric thickness of the low-index X 1 − r2 X tunnel layers is obtained from 1 by X ¼ 01ν 1 : ð Þ 2 2 3 r − X1 − X 01ν d =λ ¼ ln 1 : ð Þ 1 2 2 1=2 8 In Eqs. (1) and (3) X1 and X2 are exponential ð4πn1ÞðN − sin ϕ0Þ functions of film thicknesses given by Figure 2 shows a family of curves of the normalized Xi ¼ exp½−j4πðnidi=λÞ cos ϕi; i ¼ 1; 2: ð4Þ center-layer phase thickness θ2=π as a function of the tunnel-layer thickness parameter X1 in the range 0 ≤ In Eq. (4), ϕi is the angle of refraction in the ith layer, X1 ≤ 1 as calculated from Eq. (6) for discrete values of and λ is the vacuum wavelength of light. the interface reflection phase shift δ ¼ qπ=8, q ¼ Since FTIR takes place at the 01 interface at ϕ0, 0; 1; 2; …; 8 as a parameter. Limiting cases are iden- the light field is evanescent in the low-index tunnel tified from Eq. (6) and Fig. 2 as follows. layers, cos ϕ1 is pure imaginary, and X1 is pure real in the range 0 ≤ X1 ≤ 1. Also, because n2 ¼ n0, the angle 1. When δ ¼ 0, θ2 ¼ 0; and when δ ¼ π, θ2 ¼ 2π of refraction in the high-index center layer ϕ2 ¼ ϕ0 independent of X1. These limiting values of δ occur and X2 is a pure phase factor, i.e., jX2j¼1. at the critical angle and grazing incidence [13], re- To further clarify the thickness constraint for zero spectively, and are of little or no practical interest. reflection of the ν polarization, we substitute 2. In the limit as X1 → 0, i.e., for tunnel-layer thickness of the order of a wavelength or greater, r01ν ¼ expðjδÞ; X2 ¼ expð−jθ2Þ; ð5Þ θ2 → 2δ. Such a condition is desirable for achieving high reflectance of the unextinguished orthogonal in Eq. (3) and solve for θ2 in terms of δ, X1: polarization as discussed in Section 3. 3. When δ ¼ π=2 (01-interface reflection phase θ2 ¼ 2δ þ 2 arg½1 − X1 expð−j2δÞ: ð6Þ shift is quarter-wave), θ2 ¼ π (center layer is of quarter-wave optical thickness at oblique incidence) inde- Equation (6) is significant in that it is equally valid pendent of X1. This interesting special case provides for the p and s polarizations (for simplicity the sub- the basis of the unique PBS cube design described in script ν is dropped from δ) and depends only impli- Section 4.

1 July 2011 / Vol. 50, No. 19 / APPLIED OPTICS 3317 Fig. 2. (Color online) Normalized center-layer phase thickness θ2=π is plotted as a function of the tunnel-layer thickness parameter X1 using Eq. (6) at discrete values of the 01-interface reflection phase shift δ ¼ qπ=8, q ¼ 0; 1; 2; …; 8 such that zero reflection is achieved for the p or s polarization.

3. Embedded Symmetric Trilayer with Index-Matched layer stack with index-matched center layer n2 ¼ n0 Quarter-Wave-Thick Center Layer: Reflectance of the that is embedded in a cube prism of refractive index Orthogonal Polarization n0. The 01-interface reflection phase shift δp ¼ π=2 is For an embedded symmetric trilayer stack with achieved at ϕ0 ¼ 45° if the index ratio N is selected index-matched quarter-wave center layer (i.e., n2 ¼ such that [13] n0, θ2 ¼ π, X2 ¼ −1) the complex-amplitude reflection 2 2 4 coefficient for the orthogonal polarization (o) is given sin ϕ0 ¼ 1=2 ¼ðN þ 1Þ=ðN þ 1Þ: ð11Þ by Equation (11) yields ð1 − X Þ2 ðjδ Þ R ¼ 1 exp o : ð Þ o 2 9 pffiffiffi 1=2 1 − 2X1 sec δo expðjδoÞþX1 expðj2δoÞ N ¼ n0=n1 ¼ 2 þ 1 ¼ 1:55377: ð12Þ In Eq. (9) δo is the 01-interface reflection phase shift for the orthogonal polarization. The associated N ¼ 1:55377 intensity reflectance is obtained from Eq. (9)as It is interesting to note that causes maximum separation between the critical angle ð1 − X Þ4 δ − δ jR j2 ¼ 1 : ð Þ and the angle at which p s is maximum [13]. o 4 2 2 10 Table 1 lists some prism and film refractive indices ð1 − X1Þ þ 4ðsec δo − cos δoÞ X1 that satisfy Eq. (12). For a given prism index n0,the jR j2 Figure 3 shows a family of curves of o as a func- refractive index n1 of the tunnel layers is calculated tion of X1 in the range 0 ≤ X1 ≤ 0:1 calculated from from Eq. (12) and rounded to three decimal places. Eq. (10) at selected values of δo ¼ qπ=8, q ¼ As already noted in Section 2, zero reflection for 0; 1; 2; 3; 5; 6; 7; 8 δ ¼ π=2 . In Fig. 3 o is excluded as the p polarization is achieved when δp ¼ π=2 and jR j2 ¼ 0 it renders o , which is not physically accepta- θ2 ¼ π (center layer of quarter-wave optical thick- ble. Furthermore, δo ¼ 0; π correspond to total reflec- ness) independent of the thickness of the symmetric 2 tion, jRoj ¼ 1, at the critical angle and grazing tunnel layers. incidence, respectively, and are not of interest. As a specific design, consider an IR-transparent ZnS–Si–ZnS trilayer that is embedded in a Si cubical 4. PBS Cube Using Embedded Symmetric Trilayer prism with refractive indices n0 ¼ n2 ¼ 3:4, n1 ¼ with Index-Matched, Quarter-Wave-Thick Center Layer 2:188. Substitution of n0 ¼ 3:4, ϕ0 ¼ 45°, θ2 ¼ π in The p polarization can be totally suppressed in reflec- Eq. (7) gives the required metric thickness of the tion (hence is fully transmitted) by a symmetric tri- center layer

3318 APPLIED OPTICS / Vol. 50, No. 19 / 1 July 2011 2 Fig. 3. (Color online) Intensity reflectance for the orthogonal polarization jRoj is plotted as a function of X1 using Eq. (10) at selected values of δo ¼ qπ=8, q ¼ 0; 1; 2; 3; 5; 6; 7; 8.

d2 ¼ 0:10399λ: ð13Þ From Eqs. (16)and(17) the total thickness of the trilayer stack, To calculate the reflectance for the orthogonal (o ≡ s) polarization, Abelès’s condition [14]atϕ0 ¼ 45°is dtot ¼ d2 þ 2d1 ¼ 0:46717λ: ð18Þ used to obtain For a CO2-laser wavelength λ ¼ 10:6 μm, dtot ¼ 4:952 μ δo ¼ δs ¼ δp=2 ¼ π=4: ð15Þ m. The design presented in this section is intended for Substitution of δo ¼ π=4 in Eq. (10) gives operation at a single (e.g., laser) wavelength or with- in a narrow bandwidth centered at a given λ. This is 4 obvious from Eqs. (13), (17), and (18), which show 2 ð1 − X1Þ jRsj ¼ : ð16Þ ð1 − X Þ4 þ 2X2 that the required layer thicknesses are specific frac- 1 1 tions of the wavelength of incident light. When a col- limated laser source is used, the limited field of view From Eq. (16) it is apparent that the reflectance for s of the PBS is not a significant concern. As is well the polarization can be made as high as possible, known, the design of broadband and wide-field-of- without affecting the condition of zero reflection view PBS requires the use of a large number of thin p for the polarization, by increasing the thickness of films (see, e.g., [8,9,15]). the tunnel layers to make X1 as small as one wishes. For example, if X1 ¼ 0:001 is substituted in Eq. (16), 5. Summary jR j2 ¼ 0:999998 s is obtained. The metric thickness of Considerable simplification is achieved in the design the tunnel layerspffiffiffi is calculated by substituting of PBSs that employ symmetric LHL trilayers em- X ¼ 0:001 N ¼ð 2 þ 1Þ1=2 n ¼ 2:188 ϕ ¼ 45 1 , , 1 , 0 ° bedded in a high-index prism when the refractive in- in Eq. (8); this gives dices of the high-index center layer and the prism are matched. The p polarization is suppressed in reflec- d1 ¼ 0:18159λ: ð17Þ tion independent of the thicknesses of the tunnel

a Table 1. Selected Prism and Film Refractive Indices That Satisfy Eq. (12)

n0 (prism) 1.55377 (glass) 2.2 (ZnS) 2.35 (ZnS) 3.4 (Si) 4 (Ge) n1 (thin film) 1 1.416 1.512 2.188 2.574 a Refractive indices of ZnS listed in the third and fourth columns are appropriate to the IR and visible spectrum, respectively.

1 July 2011 / Vol. 50, No. 19 / APPLIED OPTICS 3319 layers at a 45° angle of incidence (PBS cube) by using 7. J. Mouchart, J. Begel, and E. Duda, “Modified MacNeille cube an index-matched center layer of quarter-wave thick- polarizer for a wide angular field,” Appl. Opt. 28, 2847–2853 ness. High reflectance of the s polarization is ob- (1989). “ tained by suitable choice of the thickness of the 8. L. Li and J. A. Dobrowolski, Visible broadband, wide-angle, thin-film multilayer polarizing beam splitter,” Appl. Opt. 35, tunnel layers. One such Si-cube PBS is described 2221–2225 (1996). that uses an embedded ZnS–Si–ZnS stack in the IR. 9. L. Li and J. A. Dobrowolski, “High-performance thin film I am pleased to thank Ryan Adams for assistance polarizing beam splitter operating at angles greater than the critical angle,” Appl. Opt. 39, 2754–2771 (2000). with the figures. 10. B. V. Blanckenhagen, “Practical layer design for polarizing beam-splitter cubes,” Appl. Opt. 45, 1539–1543 (2006). References 11. R. M. A. Azzam and S. R. Perla, “Polarizing properties of 1. S. M. MacNeille, “Beam splitter,” U. S. patent 2,403,731 (9 embedded symmetric trilayer stacks under conditions of frustrated total internal reflection,” Appl. Opt. 45, 1650–1656 July 1946). 2. M. Banning, “Practical methods of making and using multi- (2006). “ ” 46 layer filters,” J. Opt. Soc. Am. 37, 792–795 (1947). 12. R. M. A. Azzam and S. R. Perla, Errata, Appl. Opt. , 431–433 (2007). 3. R. P. Netterfield, “Practical thin-film polarizing beam split- “ ters,” J. Mod. Opt. 24,69–79 (1977). 13. R. M. A. Azzam, Phase shifts that accompany total internal reflection at a dielectric-dielectric interface,” J. Opt. Soc. Am. 4. D. Lees and P. Baumeister, “Versatile frustrated-total- A 21, 1559–1563 (2004). reflection polarizer for the infrared,” Opt. Lett. 4,66–67 “ ” (1979). 14. F. Abelès, Un théoreme relatif à la réflexion métallique, C. R. Hebd. Seances Acad. Sci. 230, 1942–1943 (1950). 5. R. M. A. Azzam, “Polarizing beam splitters for infrared and “ millimeter waves using single-layer-coated dielectric slab or 15. S. R. Perla and R. M. A. Azzam, Wide-angle, high- extinction-ratio, infrared polarizing beam splitters using unbacked films,” Appl. Opt. 25, 4225–4227 (1986). frustrated total internal reflection by an embedded 6. M. Gilo and K. Rabinovitch, “Design parameters of thin-film ” 46 – cubic-type polarizers for high power lasers,” Appl. Opt. 26, centro-symmetric multilayer, Appl. Opt. , 4604 4612 (2007). 2518–2521 (1987).

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